On the Smallest $(1,2)$-Eulerian Weight of Graphs

Abstract

A weight $w:E(G)\to \{1,2\}$ is called a $(1,2)$-eulerian weight of graph $G$ if the total weight of each edge-cut is even. A $(1,2)$-eulerian weight $w$ of $G$ is called smallest if the total weight $w$ of $G$ is minimum. In this note, we prove that if graph $G$ is $2$-connected and simple, and $w_0$ is a smallest $(1,2)$-eulerian weight, then either $|E_{w_0=\text{even}}|\le |V(G)|–3$ or $G=K_4$.